Invariant domains in the Hardy space over the unit disk

Inspired by some recent developments in operator theory on vector-valued function spaces, we identify and study a class of linear manifolds called invariant domains in Hilbert spaces, which implements a natural generalization of invariant subspaces for bounded operators. For coordinate multiplication operator on the Hardy space over the unit disc, a Beurling-type characterization of invariant domains together with some algebraic properties is obtained. We also go beyond the Hardy space with some related discussions on the structure of graph invariant subspaces (GIS).